The Olympics is coming to London, and the British Olympic Association (BOA) team has decreed that there will be a "Team GB" taking to the football field. They insist this has absolutely nothing to do with its 1.7m unsold tickets, which went on sale this morning (mostly for football).

The BOA evidently thought it could announce the existence of the new team and so it would come to pass. But its "historic agreement" with the Scots, Welsh and Northern Irish FAs turned out to be a sham, immediately exposed by the joint statement of the three national associations.

The Olympics isn't really about football, so you could say that none of this really matters. Sport at the Olympics is about athletes at the peak of their talents competing against the best in the world. Except in football – David Beckham, and the odd Scot and Welshman don't really set the heather afire. And yet the Team GB episode does matter, for it is about much more than football.

The English FA has apparently chosen to believe Fifa's promise that a Team GB will not have any impact on the four nations' independence. This, despite of all that Sepp Blatter and his Fifa cronies have done to humiliate England and the game of football. To the world of Fifa, the four nations of the United Kingdom has always been an anomaly to live with for the time being, but if the opportunity were to offer itself, rectify and rationalise.

The existence of a Team GB isn't just a threat to the Scots, Welsh and Northern Irish and their place on the global stage of world football, but the English, too. This could be a threat to all of us, our separate histories, traditions and teams, and for what? It poses an existential threat to one of the great emotional rallying cries of Scots national identity: the national football team. This, to some Scots, is even more serious than Margaret Thatcher and the poll tax.

The Team GB project is one without tradition, pride or purpose. The story of the four associations of the UK is a unique one; the story of the first national associations in the world, of the people who invented "association football", the Scots and English, and our parallel, interconnected histories. There is nothing in the barren world of Team GB comparable: no English highs of 1966 or lows of failing to qualify for the 1974 and 1978 World Cups, or, for the Scots, the Archie Gemmill 1978 goal or Dave Narey's "toe poke" against Brazil in 1982. The Northern Irish and the Welsh have their stories, too.

With a mixture of arrogance and rationalism, some English football fans have pointed out that the days of Scotland, Wales and Northern Ireland qualifying for major tournaments is long gone. We would all be better competing under the "Team GB" banner. But this is irrelevant: football fans care too much to accept an approach that denies the emotions and romance of their team.

What does all this say about the relationship between the UK and England? The impression is that British authorities are still enamoured with the idea of the UK as the "Great British power project": a post-imperial narrative about this island's reach, influence, and influence on Anglo-America, the Commonwealth and Europe. This is a story that is getting more difficult to fathom by the day: not only are the Scots rebelling, but also the Welsh and Northern Irish have ideas of their own.

A side effect of the Britishness project is that the British political classes – from Tony Blair and Gordon Brown to the Cameron-era Conservatives – have prevented national occasions, celebrations and events from the Olympics to the royal wedding being seen as having an English dimension. Still to this day the political establishment seem to get nervous about the St George's cross at public events – international football matches are honourable exceptions.

The union, which once took great care in understanding the traditions and histories of difference of the four nations, has instead begun to be about assimilation, standardisation and centralisation. Scotland, Wales and Northern Ireland have become forgotten entities, far away from the political centre, but the most misunderstood part is England: the last part of the UK that is denied recognition, let alone the chance to find its own democratic expression.

Everything about the balance between Englishness and Britishness, which used to be part of the strength of the union and Tory unionism, has now become a minefield to be negotiated.

The Team GB farce shows the growing ineptitude of the British establishment to keep the Britishness project on the road. For that we should be grateful, and learn to have the confidence to tell the stories of four very different, but connected nations of these isles.